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How could I minimize the following? $\vec{x} \in \mathbb{R}^8$

$$f(\vec{x}) = 3\frac{|x_1 - x_2||x_3 - x_4| + |x_5 - x_6||x_7 - x_8|}{|(x_1 - x_2)(x_3 - x_4) - (x_5 - x_6)(x_7 - x_8)|} + 1$$

there are no constraints in the variables.

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Set $u = (x_1 - x_2)(x_3-x_4)$, $v = (x_5 - x_6)(x_7-x_8)$. The function $f$ then takes the form

$$ f(u,v) = 3\frac{|u| + |v|}{|u-v|} + 1$$

And is defined on $\mathbb{R}^{2}\setminus\{0\}$. Due to the triangle inequality, we have $$|u| + |v| \geq |u-v|$$ where equality holds if and only if $u$ and $v$ have opposite signs. Thus we have $f(\vec x) \geq 4$, and this minimum is attained at all $(u,v)\neq 0 $ such that $u = -v$.

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  • $\begingroup$ Thank you, was there a way to derive the lower bound using some optimization method, like sub-gradient? $\endgroup$ Commented Jan 28, 2016 at 15:25
  • $\begingroup$ @lukkio I dont know, I have no special knowledge about optimization methods, but why would you want to do that? I don't think that it gets much easier than this. $\endgroup$
    – user159517
    Commented Jan 28, 2016 at 15:34
  • $\begingroup$ I was just curious. My main problem was to find a lower bound, since there's no upper bound. $\endgroup$ Commented Jan 28, 2016 at 15:36

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